GD 2009: Graph Drawing pp 207-218

# Manhattan-Geodesic Embedding of Planar Graphs

• Bastian Katz
• Marcus Krug
• Ignaz Rutter
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

## Abstract

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is $$\mathcal{NP}$$-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is $$\mathcal{NP}$$-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

## Keywords

Planar Graph Hamiltonian Cycle Hamiltonian Path Point Correspondence Left Endpoint
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorithms Appl. 10(2), 353–363 (2006)
2. 2.
Demaine, E.: Simple polygonizations (2007), http://erikdemaine.org/polygonization/ (Accessed May 30, 2009)
3. 3.
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)
4. 4.
Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C.-S., Spillner, A., Wolff, A.: Untangling a planar graph. Discrete Comput. Geom (2009), http://dx.doi.org/10.1007/s00454-008-9130-6
5. 5.
6. 6.
Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-geodesic point-set embeddability and polygonization. Technical Report 2009-17, Universität Karlsruhe (2009), http://digbib.ubka.uni-karlsruhe.de/volltexte/1000012949
7. 7.
Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)
8. 8.
Liu, Y., Marchioro, P., Petreschi, R., Simeone, B.: Theoretical results on at most 1-bend embeddability of graphs. Acta Math. Appl. Sinica (English Ser.) 8(2), 188–192 (1992)
9. 9.
O’Rourke, J.: Uniqueness of orthogonal connect-the-dots. In: Toussaint, G. (ed.) Computational Morphology, pp. 97–104. North-Holland, Amsterdam (1988)Google Scholar
10. 10.
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph. Combinator. 17(4), 717–728 (2001)
11. 11.
Raghavan, R., Cohoon, J., Sahni, S.: Single bend wiring. J. Algorithms 7(2), 232–257 (1986)
12. 12.
Rappaport, D.: On the complexity of computing orthogonal polygons from a set of points. Technical Report SOCS-86.9, McGill University, Montréal (1986)Google Scholar
13. 13.
Rendl, F., Woeginger, G.: Reconstructing sets of orthogonal line segments in the plane. Discrete Math 119(1-3), 167–174 (1993)
14. 14.
Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Symp. on Discrete Algorithms (SODA 1990), pp. 138–148 (1990)Google Scholar
15. 15.
Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)

## Authors and Affiliations

• Bastian Katz
• 1
• Marcus Krug
• 1
• Ignaz Rutter
• 1
• Alexander Wolff
• 2
1. 1.Faculty of InformaticsUniversität Karlsruhe (TH), KITGermany
2. 2.Institut für InformatikUniversität WürzburgGermany